
// Make newform 8775.2.a.br in Magma, downloaded from the LMFDB on 08 June 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_8775_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_8775_2_a_br();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_8775_2_a_br();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [3, 3, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, 0, 1, 0], [2, -4, -1, 1]];
    Rf_basisdens := [1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_8775_a();"
function MakeCharacter_8775_a()
    N := 8775;
    order := 1;
    char_gens := [326, 352, 8101];
    v := [1, 1, 1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_8775_a_Hecke( : Kf := Kf)
    return MakeCharacter_8775_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..prec]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -1, 0, -1], [0, -1, 1, 0], [-1, 0, 0, 0], [1, 0, 0, 1], [-2, 2, -2, -1], [2, 0, 1, -1], [-4, 1, -1, -1], [-1, 0, -1, 0], [2, -2, 1, 1], [-3, -1, -1, 0], [1, 0, 0, 3], [2, -1, -4, -1], [4, -2, -1, -2], [-3, 1, -2, -3], [-3, 1, -2, 1], [-2, 3, 3, 3], [2, -3, 2, 2], [-5, 3, -4, 0], [-3, 1, 4, 1], [-1, 2, -3, 2], [-2, 3, 3, 1], [-4, 1, 4, 1], [-7, -2, 0, 2], [-4, 4, -3, -2], [1, -1, 3, 4], [-3, 1, 4, 4], [0, 5, -2, 0], [3, -1, 1, 2], [-3, -3, 0, 0], [6, -4, -4, -3], [-2, -7, 3, 2], [-1, 3, 0, -1], [3, -8, 0, 4], [-8, 9, -2, -3], [-6, -1, -4, 2], [-11, 2, -2, 4], [13, -3, -1, 1], [-6, 5, -7, 0], [0, -2, -2, -5], [-10, 0, -2, 5], [0, 8, -1, 2], [7, -4, 6, -2], [-11, 2, -6, -1], [-3, -11, 2, -2], [3, -4, 9, 2], [1, -3, 4, -2], [-3, -5, -2, 1], [8, -1, 6, -1], [5, 5, -2, 5], [-3, 1, 1, -2], [1, -4, 7, -2], [-2, -7, 0, -8], [-3, -2, 7, -3], [0, 7, -1, 6], [-1, 3, 2, 6], [-2, -12, 3, -6], [-8, -8, 3, -2], [11, -5, 0, -8], [-10, 13, -3, -4], [3, 5, 1, -7], [0, 3, -6, -3], [4, -6, -3, -3], [-6, 6, 3, 9], [0, 7, 5, 1], [-18, -1, -1, -1], [10, -9, 0, 4], [1, 5, 4, -4], [18, -8, 11, 6], [-8, 7, 1, -5], [-7, 10, -1, 4], [-1, 7, 0, -2], [-3, 1, 6, 1], [-12, 5, -1, 6], [4, -5, 14, 1], [-6, -1, -5, 5], [13, 6, 1, 7], [-2, 14, 0, -1], [14, 4, -5, 2], [-2, 2, -5, -4], [9, -7, -3, 0], [0, -4, 0, 2], [-2, 11, 2, -4], [2, -10, -3, -1], [-7, 6, -13, -4], [-13, -11, 5, -2], [-3, -11, 1, 3], [5, 1, 11, 10], [9, 9, -2, 6], [23, 1, 4, 2], [3, -4, -1, 5], [3, 8, -11, -3], [5, 6, 5, -3], [-14, 12, 1, 1], [-1, -2, 0, -1], [-7, -10, 11, 2], [-15, 8, -10, -10], [19, -10, 4, 5], [11, -17, 4, 7], [21, -2, 4, -12], [1, 6, -1, -11], [-9, -10, -5, -12], [-13, 12, 9, 3], [12, 5, -3, -4], [-14, 8, -3, -5], [10, 4, -10, -2], [-11, 10, 4, 10], [-10, -3, 8, 0], [-20, 12, -3, 0], [3, -7, -3, -16], [3, 2, 15, 6], [-8, -19, 6, -4], [25, -1, 5, 5], [-15, -2, -8, -12], [-3, -10, -1, 2], [-6, -2, -14, -9], [-7, 6, 8, 2], [0, -2, -2, -3], [-10, -9, -3, 6], [-8, -12, 13, -3], [26, -4, 3, 5], [7, 12, 2, 4], [-6, -8, 9, 1], [-26, 1, -3, 4], [-24, 10, -7, -8], [-19, 15, 2, 2], [-2, 3, -7, -9], [14, -5, 7, 4], [13, -7, 10, 2], [6, 1, -8, -12], [-20, -1, -1, 5], [6, 8, 3, 5], [10, 17, -1, -2], [13, 2, -7, -10], [20, -20, 6, 8], [18, -10, 19, 8], [23, 1, 10, 8], [3, -13, 14, -3], [-3, 4, -16, 1], [-3, -7, 16, 3], [-11, 3, -13, 0], [5, 11, -2, -7], [-6, 4, 17, 1], [14, 0, 4, 5], [-11, 15, -16, -6], [-19, 11, -7, 5], [9, -20, -4, -2], [-8, -14, 7, -8], [3, -22, 2, -7], [-18, 8, 1, -6], [-14, 3, -7, 3], [-33, 1, -5, -3], [20, -8, 5, 1], [10, -2, 6, 13], [-5, 17, 1, -7], [-9, 7, -5, 9], [5, -2, -10, -2], [3, 6, -4, -6], [-27, 1, -5, -3], [30, 11, -3, 3], [12, 5, -8, 5], [-11, 14, -3, -5], [-12, 7, -20, -9], [18, -15, 24, 9], [11, -3, -2, 0], [-23, 9, -12, -6], [5, -12, 12, -6], [-5, 7, -10, 4], [27, 12, -5, 3], [-12, 1, 5, -2], [4, 8, 15, 16], [-31, 7, -6, 7], [18, 7, -8, -8], [-3, 7, 10, 9], [-9, 10, -22, -2], [-42, 0, -5, 0], [-9, -1, -1, -1], [-28, 3, -16, 12], [11, -2, 3, -8], [33, 6, 0, 0], [-11, 12, -20, -3], [-14, 4, 5, 7], [-7, 0, 23, 2], [-21, -8, -4, 15], [26, -11, 13, 1], [-10, 4, -12, -20], [26, -15, 15, 15], [28, 5, -2, 1], [-14, -18, 3, -6], [-9, -4, 16, 6], [6, 19, -9, 1], [0, -1, -19, -15], [9, 14, -4, -3], [-22, 12, 3, -1], [-23, 5, 2, 8], [-8, -6, -6, 15], [-5, 1, -16, -8], [26, -22, -1, 2], [-6, 10, -4, -3], [0, 4, 6, -8], [-12, 11, 6, 2], [-27, 13, -18, -8], [-8, 12, 2, 13], [21, 4, -8, 0], [8, -3, 6, 6], [25, 15, -15, 1], [14, -5, -15, -7], [-17, -13, 7, 2], [13, -12, -5, 6], [13, -23, -1, -2], [-25, -2, -13, 4], [-8, -9, 4, 4], [12, -8, -8, -3], [2, -18, -3, -6], [22, 6, -7, -12], [-2, 17, -15, 4], [-16, 0, 11, 5], [7, -8, -1, 1], [-1, 0, 18, -3], [-14, 26, -18, -7], [-1, 2, -4, -10], [-8, -21, 2, -9], [-26, -19, 5, -5], [-12, -11, 2, -11], [-2, 20, 0, -8], [1, 17, -12, -11], [0, 14, 1, 11], [-5, 1, -8, -8], [15, 2, 11, -1], [11, -18, 3, -18], [-47, -4, 7, 2], [1, 1, -1, 4], [-32, -9, 4, 6], [-17, -14, 6, -8], [0, -5, -5, -2], [23, -12, -1, -10], [-22, 15, -15, -16], [0, 2, -18, -15], [-29, 5, -18, 7], [13, 5, 3, 14], [-7, 7, -11, -8], [5, 0, -18, -12], [-19, -23, 1, -4], [33, -9, -9, -6], [13, 3, -26, -3], [50, -5, 15, 11], [30, 13, 5, -11], [-12, 16, -12, -15], [-1, 25, -6, 1], [37, -12, 20, 13], [0, 18, -7, 6], [5, 9, 3, 6], [11, -31, 6, 11], [42, -8, 5, 0], [7, 5, -7, 2], [10, 9, 3, -3], [-5, -2, -7, -5], [-13, 7, -25, -11], [-32, 15, 10, 3], [-56, 2, 3, 4], [20, -12, 10, 3], [-5, -9, 9, 3], [-26, -27, 14, -2], [-4, -9, -19, -6], [-26, 8, -6, 13], [35, -7, -3, -4], [-16, 13, -9, -8], [12, 8, -6, 12], [-27, 10, -12, -17], [37, 18, 1, 12], [-22, 7, -13, -14], [45, -5, 5, -2], [-12, -10, -11, 14], [3, 32, 0, 2], [25, 6, 10, -11], [-9, -11, -3, 7], [5, -12, -6, -18], [-2, 17, 2, -2], [-10, -21, 6, 2], [-26, -19, 13, -4], [-39, -9, -9, 0], [-7, -18, 4, 18], [8, -14, -19, 1], [12, -24, 20, -3], [-2, 9, -8, -6], [3, 3, -5, 0], [15, -29, -11, -5], [-12, 22, -6, -6], [-34, 1, 6, -1], [-26, 3, 10, 13], [4, -16, 5, 13], [-3, 4, -5, -9], [-3, -1, 11, 11], [31, -14, -19, -11], [8, 18, -9, 3], [36, 7, 7, 0], [-12, -22, 22, 3], [43, 15, -8, 3], [10, 15, 8, 9], [17, -10, 6, 17], [26, -6, -4, -3], [21, -27, 31, 9], [-2, 2, -3, -1], [-26, 18, -15, -12], [-12, -30, 11, -3], [-33, 13, 0, -2], [16, -6, -8, -2], [17, -14, 3, 7], [11, 25, -21, 5], [10, 18, -9, 13], [-29, 1, 1, 16], [-27, -16, 9, -4], [-16, -2, 9, -1], [11, -3, 22, 6], [17, 12, 8, -19], [-54, 3, 11, 9], [-21, 14, 1, -13], [46, -3, 16, 16], [10, -13, 21, 14], [13, 9, 10, 12], [-6, 16, -23, -6], [15, 2, 2, 14], [13, -26, -6, -14], [-15, 10, -16, 10], [17, -9, 19, -15], [-4, -2, 8, -5], [38, -15, 20, 23], [41, -18, 11, 8], [12, 1, 4, -14], [-3, 9, -29, -15], [6, 13, -5, 19], [-10, 28, -9, -10], [-52, -9, 10, 0], [-15, -21, 0, -15], [-28, -9, -10, 24], [-17, -34, 10, 7], [-13, 36, -1, 0], [-11, -9, -14, -12], [-1, -11, 18, 1], [17, -29, -9, 8], [-18, 27, -1, 12], [7, -4, 7, -10], [63, -7, 2, -3], [4, 19, -7, -17], [26, -15, -5, -15], [30, 2, 7, 14], [7, -24, 1, -26], [16, 30, -5, -5], [-10, 24, -7, 0], [-13, 16, -3, 1], [-9, -25, -8, 0], [-45, 23, -16, -13], [17, 18, -3, -15], [-34, 20, 8, 14], [8, -6, 16, 17], [-14, -1, -8, -11], [-23, 10, -5, 7], [-7, 14, 1, 2], [-15, 11, -1, 14], [-22, 7, 15, 8], [28, -16, -7, -2], [-33, -13, -7, 3], [38, -14, 9, -14], [4, -12, 8, 15], [3, -7, 19, -3], [-21, 23, 4, 17], [-3, -2, 28, 1], [-13, -3, -4, -22], [29, -27, -15, -6], [-2, -17, -11, 4], [-19, -28, -2, -7], [-65, 20, -14, -16], [6, -34, 22, -1], [-18, -3, 27, 9], [15, 7, 6, -12], [-19, 4, -4, -5], [28, -1, -8, -5], [-24, -19, 14, -1], [-30, 0, 0, -12], [-36, 8, -21, -18], [39, -7, 2, -22], [-5, 1, -23, -5], [43, 20, 2, 5], [-16, -21, 0, 23], [-6, -26, -8, -20], [-47, -21, 1, -6], [-8, 6, -10, 0], [42, 5, -19, -3], [9, -8, -6, -17], [43, -10, 12, -8], [10, 19, 18, 16], [-7, 1, -3, 7], [-30, 16, 11, 0], [10, -18, 15, 12], [10, -19, 6, 20], [-4, -16, 2, 11], [19, -10, -20, -16], [13, 0, 23, 13], [26, -3, -13, 0], [28, 5, 5, -5], [13, -12, 6, -6], [-18, 9, 16, 24], [12, 16, 3, -3], [5, 6, 11, -18], [-34, 10, 23, 1], [-1, -9, 22, 8], [-23, -18, 0, -9], [25, -12, -11, -5], [-40, -11, -3, -20], [65, -12, -7, -1], [17, 19, 12, -7], [-32, 2, -9, -19], [-16, -11, 2, -2], [-7, -31, 6, 8], [-1, -42, 17, 5], [36, 2, -4, -1], [8, 0, -13, -7], [5, -8, -9, -7], [-51, -13, -5, -15], [-17, 18, -23, -6], [31, -4, 13, 19], [-6, -27, -9, -15], [-11, -6, 6, 18], [14, 22, -21, 5], [13, -32, 33, 7], [23, -22, -24, -22], [-31, 6, -7, 12], [-15, -32, 6, -9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8775_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8775_2_a_br();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_8775_2_a_br(:prec:=4)
    chi := MakeCharacter_8775_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_8775_2_a_br();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_8775_2_a_br( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8775_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, 3, -5, -1, 1]>,<7,R![33, -12, -14, 1, 1]>,<17,R![3, 6, -5, -4, 1]>,<19,R![-137, -146, -33, 4, 1]>],Snew);
    return Vf;
end function;